I use kayaks pretty exclusively, so feel free to ask me any questions.
KayaksIts very much a personal choice. I would probably recommend a Sit in Kayak for you wife, unless she will be around rough water. You sit low to the water and as long as it the bottom is wide its pretty stable. Stable in a kayak is determined by the flatness and width of the bottom. Wide is more stable and slower and less wide and longer and you get a faster, less stable kayak. A similar kayak would be a Sea Kayak, such as a Wilderness Systems tsunami.
Wilderness Systems tsunami (Sea Kayak) notice long and skinny
Wilderness Commander, My favorite hunting and fishing kayak. It's kind a canoe/kayak hybrid and have a lower seat like a normal kayak and an upper seat where you can fold the lower seat under and sit in the depression behind the seat. This is very comfortable and lets you bend you legs more. I sit sideways on the upper seat fishing in the summer with my legs in the water. This one is very stable. This one is not mine, mine is camo, of course. You can stand in this one also, but the sides rub your calf if you do it a lot.
Here is another Sit in Kayak my daughter uses all the time. Its a Old Town Vapor 10. Only 10' long but very wide and stable. We got this for $200 used from a place that used it as a rental. This was her a couple of years back and you can see she could sit sideways in it. It's all about learning your balance.
My rough water and dog kayak is the Wilderness Ride 135, which is a Sit on Top kayak. I can take it through the surf and into the Ocean if I wanted too. You can't sink it unless you drill a hole in it. I have had this out in water that looked like the guys near me in Jon boats looked like they were going to get sick.
I use it on the Chesapeake for stripers and fish the shipping channels and take all the wakes the boats throw. Mine is camo, though for Hunting and I stand in this all the time.
BalanceStaying upright is a kayak is really simple and most people never understand it and fight it. Your butt is always in the seat. The key is where your shoulders go, so does the kayak. If the kayak goes too far left, shift your shoulders right. It takes gettign used too, but after a trip or two it's automatic.
PaddlesDon't buy the crappy paddles they sell at Cabelas or some other stores, they suck! Standard blades are just that and not veyr helpful. A cheap Carlisle Day tripper can be bought from Dick's or a place like that for $60 and they have a defined top and bottom to them and a semi cupped side. The aluminum shafts can get worn out if you are a hard paddler, as I am. These would serve most kayakers plenty enough as they don't use them enough or paddle as hard.
Personally I like the Bending Branches Classic Angler. A combination of price and is very durable. I have the Green one.
http://www.kayakfishinggear.com/collections/paddles/products/bending-branches-angler-classicYes, its $140, but it has a fiberglass shaft for stiffness and longevity and ABS blades. I fish the Upper Potomac and use it as a push pole on the rocks to get through rapids. Fiberglass blades don't take to rocks very well and Carbon fiber shafts reduce weight but are so expensive. I have one for the bay and it's nice, but it's so light if I anchor on the bay the wind can literally blow it off the kayak. I don't have that problem with my Classic Angler paddle